caa a22 4500 60551514X CHVBK 20210128100707.0 cr unu---uuuuu 210128e20150501xx s 000 0 eng 10.1007/s00205-014-0811-4 doi (NATIONALLICENCE)springer-10.1007/s00205-014-0811-4 Localized Concentration of Semi-Classical States for Nonlinear Dirac Equations [Elektronische Daten] [Yanheng Ding, Tian Xu] The present paper studies concentration phenomena of the semiclassical approximation of a massive Dirac equation with general nonlinear self-coupling: $$\begin{array}{ll}-i \hbar \alpha \cdot \nabla w+a \beta w + V (x) w = g (|w|) w.\end{array}$$ - i ħ α · ∇ w + a β w + V ( x ) w = g ( | w | ) w . Compared with some existing issues, the most interesting results obtained here are twofold: the solutions concentrating around local minima of the external potential; and the nonlinearities assumed to be either super-linear or asymptotically linear at the infinity. As a consequence one sees that, if there are k bounded domains $${\Lambda_j \subset \mathbb{R}^3}$$ Λ j ⊂ R 3 such that $${-a < \min_{\Lambda_j} V=V(x_j) < \min_{\partial \Lambda_j}V}$$ - a < min Λ j V = V ( x j ) < min ∂ Λ j V , $${x_j\in\Lambda_j}$$ x j ∈ Λ j , then the k-families of solutions $${w_\hbar^j}$$ w ħ j concentrate around x j as $${\hbar\to 0}$$ ħ → 0 , respectively. The proof relies on variational arguments: the solutions are found as critical points of an energy functional. The Dirac operator has a continuous spectrum which is not bounded from below and above, hence the energy functional is strongly indefinite. A penalization technique is developed here to obtain the desired solutions. Springer-Verlag Berlin Heidelberg, 2014 Ding Yanheng Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, China aut Xu Tian Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, China aut Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 216/2(2015-05-01), 415-447 0003-9527 216:2<415 2015 216 205 https://doi.org/10.1007/s00205-014-0811-4 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00205-014-0811-4 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Ding Yanheng Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, China aut NATIONALLICENCE 700 1- Xu Tian Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, China aut NATIONALLICENCE 773 0- Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 216/2(2015-05-01), 415-447 0003-9527 216:2<415 2015 216 205